Before Christ, there would be two tabernacles in the temple: an outer and an inner tabernacle. Priests would minister at the outer tabernacle by caring for the lamps, by burning incense to God twice daily, and by giving weekly offerings of bread. Once a year, the high priest of those priests would sacrifice an animal, and then he would dare to enter through the second veil into the inner tabernacle. There, he would sprinkle the mercy seat, the place of God's presence, with the sacrificial blood, and would then invoke the name of the Saviour God in atonement for the sins of Israel. Then he would leave.

When Christ offered up His sacrifice, He did not go to the man-made outer or inner tabernacle. He passed through into heaven itself to the seat of God the Father. There at the throne of God He sat down at the right hand of the Father as our most perfect intercessor. Our High Priest went up to that most sacred place to ask God to forgive us, and He remained there. That is how Jesus is even now serving as High Priest.

Since we no longer need high priests to intercede for us by way of repeated bloody sacrifice, why do Catholics have priests? They are obviously not re-sacrificing Christ when they say the Mass, so what do they do as priests, and why are they called priests if they are not (to put it bluntly) killing something?

(Answers from other denominations (Orthodox?) that have compatible reasons for having priests are welcome.)

@Waggers Yes, this is primarily focused on the priesthood described in Hebrews 9, because this is the one that was explicitly abolished due to Christ's role as intercessor. Helpful mention of the other priesthoods is very much welcome in an answer.
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AlypiusApr 8 '13 at 18:40

2 Answers
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The Old Testament priesthood is a foreshadow of the New Testament priesthood, but many features are different; the word for OT/pagan priest and for NT priest is also different in some languages including Greek and Latin.

The OT Levite priesthood was tied to bloodline of Aron and their role was to sacrifice animals and other offerings in the Temple. These features of priesthood were abandoned with the sacrifice of Jesus.

The ministerial priesthood (and common priesthood shared by all baptised too) is derived from the priesthood of Christ. Similarly to several levels of discipleship of Jesus' followers in the Gospels (Twelve/ Seventy/ the crowds), there are different levels of participation of Christ's priestly authority.

The ministerial priests' role is to serve people in the place of Christ, especially while performing the sacraments. As a sacrifice was a definition feature of Levite priesthood, sacrament is a crucial point of Catholic ministerial priesthood.
As there were only the Twelve in the Upper Room while Jesus said about the Eucharist: "Do it in the remembrance of me!", we believe that the Twelve and their successors (the bishops and priests) have the power to perform the visible part of the miracle of transsubstantiation (about the other Christians and this priestly power, there's nothing in the Bible, but tradition said "no").
The same is true about the confession and most other sacraments - they were given to the twelve, so only those ordained to do so have the power.

It's true enough that they are not re-sacrificing Christ. But in a mystical way the Catholic priest acts in persona Christi to offer, in an unbloody way, the very same sacrifice that was offered in a bloody way on Calvary. Here is the modern Catholic Catechism (paragraph 1367), quoting the 16th-century Council of Trent (my emphases):

The sacrifice of Christ and the sacrifice of the Eucharist are one single sacrifice: "The victim is one and the same: the same [Christ] now offers through the ministry of priests, Who then offered Himself on the cross; only the manner of offering is different."

Thus a principal point of having priests, in the Catholic Church, is to continually make the unique, unrepeatable sacrifice of Calvary mystically present to us here and now. Our priests, like Christ himself, are of the order of Melchizedek, not the order of Aaron. Indeed there is in a sense only the priesthood of Christ -- as St. Thomas Aquinas says (and the modern Catechism quotes, paragraph 1545):

@Alypius made some edits, hope they help. It would take a lengthy theological treatise to unpack it, but the basic doctrine is that Christ offered himself once and for all on Calvary but mystically makes this offering present in many times and places through the ministry of priests in the Eucharistic liturgy. A corollary of this doctrine is that the first Eucharistic liturgy occurred at the Last Supper -- and thus in a mysterious sense the Twelve were present at Calvary the night before it occurred in time. AFAIK this teaching is common to Catholics and Orthodox, btw.
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Ben DunlapApr 10 '13 at 20:59

Excellent points, especially the quote you added to the answer about priests being ministers to Christ.
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AlypiusApr 11 '13 at 5:08